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Properties

Label 2-336-336.173-c1-0-54
Degree $2$
Conductor $336$
Sign $-0.181 + 0.983i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.32 − 0.486i)2-s + (−0.388 − 1.68i)3-s + (1.52 − 1.29i)4-s + (−0.224 − 0.837i)5-s + (−1.33 − 2.05i)6-s + (2.27 − 1.35i)7-s + (1.39 − 2.45i)8-s + (−2.69 + 1.31i)9-s + (−0.705 − 1.00i)10-s + (−1.49 + 5.58i)11-s + (−2.77 − 2.07i)12-s + (−0.124 + 0.124i)13-s + (2.35 − 2.90i)14-s + (−1.32 + 0.703i)15-s + (0.659 − 3.94i)16-s + (−1.74 − 3.02i)17-s + ⋯
L(s)  = 1  + (0.938 − 0.344i)2-s + (−0.224 − 0.974i)3-s + (0.763 − 0.646i)4-s + (−0.100 − 0.374i)5-s + (−0.546 − 0.837i)6-s + (0.858 − 0.512i)7-s + (0.494 − 0.869i)8-s + (−0.899 + 0.437i)9-s + (−0.222 − 0.316i)10-s + (−0.451 + 1.68i)11-s + (−0.800 − 0.598i)12-s + (−0.0344 + 0.0344i)13-s + (0.630 − 0.776i)14-s + (−0.342 + 0.181i)15-s + (0.164 − 0.986i)16-s + (−0.423 − 0.732i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.181 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.181 + 0.983i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.181 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37927 - 1.65753i\)
\(L(\frac12)\) \(\approx\) \(1.37927 - 1.65753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 0.486i)T \)
3 \( 1 + (0.388 + 1.68i)T \)
7 \( 1 + (-2.27 + 1.35i)T \)
good5 \( 1 + (0.224 + 0.837i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.49 - 5.58i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (0.124 - 0.124i)T - 13iT^{2} \)
17 \( 1 + (1.74 + 3.02i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.346 - 1.29i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.74 - 3.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.99 + 1.99i)T - 29iT^{2} \)
31 \( 1 + (-4.42 + 2.55i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.82 - 10.5i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.85iT - 41T^{2} \)
43 \( 1 + (4.53 - 4.53i)T - 43iT^{2} \)
47 \( 1 + (5.88 - 10.1i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.64 - 6.15i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-7.87 - 2.10i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.443 - 1.65i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-0.813 + 3.03i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.67T + 71T^{2} \)
73 \( 1 + (5.43 + 9.42i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.17 + 2.04i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.05 + 3.05i)T + 83iT^{2} \)
89 \( 1 + (-6.91 - 3.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.88iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.71803252605159978001705149972, −10.67507845933637704485024603676, −9.708486545785772890260557243005, −8.065055849741833143630129647468, −7.34004806975497563570997847793, −6.43618549216425584362966412912, −5.05342458660913259749874542440, −4.48823381383030610074907530558, −2.57100009093778533322713833360, −1.39281755403883988771019954803, 2.69152556766377685716901249022, 3.74045798492528265306230475381, 4.94278587307371323531496879687, 5.68326686931812013339061779583, 6.64844353802219171886412237210, 8.266952663590036187556781248039, 8.634637217208858386100826044093, 10.36629841073634110996195533266, 11.14520420930918132940917076667, 11.53433638321298738442864011790

Graph of the $Z$-function along the critical line